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Theorem scutfo 27951
Description: The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
Assertion
Ref Expression
scutfo |s : <<s –onto No

Proof of Theorem scutfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scutf 27866 . 2 |s : <<s ⟶ No
2 lltropt 27920 . . . . 5 ( L ‘𝑥) <<s ( R ‘𝑥)
3 df-br 5170 . . . . 5 (( L ‘𝑥) <<s ( R ‘𝑥) ↔ ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s )
42, 3mpbi 230 . . . 4 ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s
5 lrcut 27950 . . . . 5 (𝑥 No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
65eqcomd 2740 . . . 4 (𝑥 No 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
7 fveq2 6919 . . . . . 6 (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩))
8 df-ov 7448 . . . . . 6 (( L ‘𝑥) |s ( R ‘𝑥)) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩)
97, 8eqtr4di 2792 . . . . 5 (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = (( L ‘𝑥) |s ( R ‘𝑥)))
109rspceeqv 3653 . . . 4 ((⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s ∧ 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥))) → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
114, 6, 10sylancr 586 . . 3 (𝑥 No → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
1211rgen 3065 . 2 𝑥 No 𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)
13 dffo3 7134 . 2 ( |s : <<s –onto No ↔ ( |s : <<s ⟶ No ∧ ∀𝑥 No 𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)))
141, 12, 13mpbir2an 710 1 |s : <<s –onto No
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2103  wral 3063  wrex 3072  cop 4654   class class class wbr 5169  wf 6568  ontowfo 6570  cfv 6572  (class class class)co 7445   No csur 27693   <<s csslt 27834   |s cscut 27836   L cleft 27893   R cright 27894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-1o 8518  df-2o 8519  df-no 27696  df-slt 27697  df-bday 27698  df-sslt 27835  df-scut 27837  df-made 27895  df-old 27896  df-left 27898  df-right 27899
This theorem is referenced by: (None)
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